Family and Subfamily of Subsets of a Set

Let $X$ be a set, let $A$ be a set, and suppose that for each element $a\in A$ a subset $U_a\subseteq X$ is specified. The collection $(U_a)_{a\in A}$ is called a family of subsets of $X$ indexed by $A$.

If $B\subseteq A$, then the collection $(U_b)_{b\in B}$ is called the subfamily of $(U_a)_{a\in A}$ indexed by $B$.